Approximation and optimality necessary conditions in relaxed stochastic control problems

Seïd Bahlali,Brahim Mezerdi,Boualem Djehiche

doi:10.1155/jamsa/2006/72762

Abstract

We consider a control problem where the state variable is a solution of a stochastic differential equation (SDE) in which the control enters both the drift and the diffusion coefficient. We study the relaxed problem for which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by an orthogonal martingale measure. Under some mild conditions on the coefficients and pathwise uniqueness, we prove that every diffusion process associated to a relaxed control is a strong limit of a sequence of diffusion processes associated to strict controls. As a consequence, we show that the strict and the relaxed control problems have the same value function and that an optimal relaxed control exists. Moreover we derive a maximum principle of the Pontriagin type, extending the well-known Peng stochastic maximum principle to the class of measure-valued controls.

Highlights

We are interested in questions of existence, approximation, and optimality of control problems of systems evolving according to the stochastic differential equation t t xt = x + b s, xs, us ds + σ s, xs, us dBs, (1.1)
The case of an stochastic differential equation (SDE) where the diffusion coefficient depends explicitly on the control variable has been solved by El-Karoui et al [5], where the optimal relaxed control is shown to be Markovian
Under a continuity condition of the coefficients and pathwise uniqueness of the controlled equations, each relaxed diffusion process is a strong limit of a sequence of diffusion processes associated with strict controls

Summary

Introduction

Under a continuity condition of the coefficients and pathwise uniqueness of the controlled equations, each relaxed diffusion process is a strong limit of a sequence of diffusion processes associated with strict controls The proof of this approximation result is based on Skorokhod’s selection theorem, a limit theorem on martingale measures and Mitoma’s theorem [20] on tightness of families of martingale measures. The second main result of this paper is a maximum principle of the Pontriagin type for relaxed controls, extending the well-known Peng stochastic maximum principle [22] to the class of measure-valued controls. This leads to necessary conditions satisfied by an optimal relaxed control, which exists under general assumptions on the coefficients.

Formulation of the problem

Approximation and existence results of relaxed controls

Findings

Maximum principle for relaxed control problems